This idea is an homage to some of the weirdness in the 3e Unearthed Arcana book, and a friendly wave to 5e D&D as well. There is probably some stuff from GURPs and WaRP in there too.
Instead of assigning ability scores, you assign a die to the various abilities, from 1d4 (not so great) to 1d12 (superhero).
One way of doing this is to start everybody out at 1d6 for everything (‘average’) and let them increase/decrease scores from there (in other words, let one ability fall to 1d4 to raise another to 1d8, etc.). You can also randomize the assigned dice with a d10 roll or something. I like to start folks out at 1d8 for two ‘main’ abilities and 1d6 for everything else, then up and down as you please. ANYWAY…
When making a check, roll your Ability Die (AD) + 2d6.
If in an ‘advantageous’ position (surprise, flanking, particular skill set, etc.) roll 3d6 + AD, drop the lowest.
If in a ‘disadvantageous’ position (being flanked, on slippery ground, drank too much) roll 3d6 + AD drop the highest UNLESS the highest is the AD, in which case you drop the next highest.
If you want to have weirdstupidfun, you can replace d20 rolls with the above system as well, and just use STR or DEX to hit based on weapon.